Formula relativization to a class

The relativization ${\phi }^M$ of a formula $\phi$ is obtained recursively by restricting every quantifier to $M$:For parameters in $M$, the ambient statement ${\phi }^M$ holds exactly when the structure $(M,\in )$ satisfies $\phi$.

A formula is absolute between transitive classes $M\subseteq N$ when it has the same truth value in both structures for parameters from $M$.

Let $M\subseteq N$ be transitive models of a theory proving both an existential-bounded and a universal-bounded equivalent of a formula. For parameters in $M$, the existential form gives upward absoluteness and the universal form gives downward absoluteness. Consequently the models agree on the formula. The equivalences must hold in both models.

For a set relation in nested transitive models of ZFC, well-foundedness is absolute. An inner-model ordinal rank function witnesses upward absoluteness, and a fixed nonempty subset without a minimal member witnesses failure in both models.

A formula with unchanged truth value between the specified models, using parameters shared by them. Usually the models in set-theoretic applications are transitive and nested; the class of models is part of the assertion of absoluteness.

For limit ordinals $\alpha \le \beta$, the structures $V_{\alpha }$ and $V_{\beta }$ agree on whether any shared set is a cardinal number. They agree on being an ordinal. If $\delta <\alpha$ is not a cardinal number, a bijection from some $\gamma <\delta$ onto $\delta$ has rank of a set at most $\delta +3$, hence belongs to $V_{\alpha }$. Its defining properties are bounded formulas in set theory. Both structures therefore see the same failure witness. This full agreement is stronger than downward absoluteness of cardinalhood between arbitrary transitive models.

A formula is upward absolute when its truth in a smaller transitive class implies its truth in a larger one. Existential formulas with bounded matrices are upward absolute because their witnesses remain available.

A formula is downward absolute when its truth in a larger transitive class implies its truth in a smaller one. Universal formulas with bounded matrices are downward absolute.

A bounded formula in set theory has every quantifier in one of the forms $\forall x\in y$ or $\exists x\in y$. Such formulas are absolute between transitive models containing their parameters.

A first-order formula is ZF-equivalent bounded if ZF proves it equivalent, with the same free variables, to a syntactically bounded formula in set theory. It need not itself have only bounded quantifiers. Between transitive models of ZF, use the bounded equivalent and the axioms in each model to prove set-theoretic absoluteness. The syntactic bounded-formula result requires only transitivity; the provable-equivalence extension additionally requires the theory used for the equivalence.

Transitive models of ZFC have the same natural numbers and agree on whether a shared set is a finite set. Consequently they also agree on whether it is an infinite set.

If a larger transitive model regards an ordinal $\kappa$ as a cardinal number, then a smaller transitive model does too: any bijection in the smaller model witnessing otherwise would remain in the larger one. The converse can fail when the larger model contains a new bijection between $\kappa$ and a smaller ordinal.

If a smaller transitive model has a function witnessing that $x$ is a countable set, the same witness exists in every larger transitive model. Downward absoluteness can fail because a larger model may contain a new enumeration of $x$.

The statement that $\kappa$ is a cardinal number with $\mathrm{cf}(\kappa )<\kappa$ is generally neither upward nor downward absolute. Upward absoluteness can fail when a larger model collapses the cardinal; downward absoluteness can fail when a larger model adds a short cofinal sequence to a cardinal that the smaller model regards as regular.

Let $M$ be a transitive set of cardinality $\kappa$ that models enough set theory and contains $\kappa$. The internal successor cardinal $\alpha =({\kappa }^{+}{)}^M$ is an ordinal in $M$, so transitivity gives $\alpha \subseteq M$ and hence $|\alpha |\le \kappa$ externally. Although $M$ regards $\alpha$ as a cardinal number, an ambient rank containing a bijection between $\kappa$ and $\alpha$ does not. Cardinalhood can therefore fail to be absolute even between transitive models.

If transitive models of enough set theory contain the same $V_{\alpha +1}$, then they agree on whether $\alpha$ is a strongly inaccessible cardinal. They have the same subsets and functions on every ordinal below $\alpha$, so they agree on cardinality, regularity, and the strong limit cardinal property.

The Lévy hierarchy classifies formulas of set theory by their alternations of unbounded quantifiers, ignoring bounded quantifiers of the forms $\forall x\in y$ and $\exists x\in y$.

A first-order formula is ${\Delta }_1$ modulo ZFC if that theory proves it equivalent, with the same free variables, both to a Sigma-one formula in set theory and to a universal unbounded quantifier block with a bounded formula in set theory as matrix. The theory used for provable equivalence matters: modulo ZF is a stronger requirement.

A syntactically ${\Pi }_1$ first-order formula has an unbounded universal quantifier block followed by a bounded formula in set theory. Universal quantification makes it downward absolute between transitive models. A Pi-one formula modulo ZF may have another syntax but has such an equivalent provable in ZF.

A first-order formula is syntactically ${\Sigma }_1$ when it is an unbounded existential quantifier block followed by a bounded formula in set theory. Equivalence provable in a specified theory gives the corresponding modulo-theory notion. Existential witnesses show upward absoluteness between transitive models containing the parameters.

A first-order formula is ${\Pi }_1^{\mathrm{ZF}}$ if ZF proves it equivalent, with the same free variables, to a universal unbounded quantifier block followed by a bounded formula in set theory. Bounded quantifiers do not increase its level in the Lévy hierarchy. Such properties are downward absolute between appropriate transitive models of ZF.

An infinite cardinal number $\kappa$ is a regular cardinal if every function from an ordinal $\delta <\kappa$ into $\kappa$ has bounded range. Write this as $\forall f\forall \delta \in \kappa (\mathrm{Map}(f,\delta ,\kappa )\Rightarrow \exists \beta \in \kappa \forall \xi \in \delta f(\xi )<\beta )$. The graph predicates are bounded formulas in set theory, and cardinalhood is Pi-one definable, so adding infinite cardinalhood yields a Pi-one formula modulo ZF.

An ordinal $\kappa$ is a cardinal number precisely when no smaller ordinal has a bijection onto it. Ordinalhood and the graph predicate for a bijection are bounded formulas in set theory, so $\mathrm{Ord}(\kappa )\wedge \forall f\forall \alpha \in \kappa \neg \mathrm{Bij}(f,\alpha ,\kappa )$ is a Pi-one formula modulo ZF. This does not assert that every set can be assigned an ordinal cardinal without the axiom of choice.

A formula is ${\Delta }_1^{\mathrm{ZF}}$ when ZF proves it equivalent both to a ${\Sigma }_1$ formula and to a ${\Pi }_1$ formula. Such formulas are absolute between suitable transitive models of finite fragments of ZF.